1
vote
0answers
27 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
2
votes
1answer
50 views

Calculating a Lebesgue integral

Calculate the Lebesgue integral of, $$\lim_{n\to\infty}\int_{[0,1]}\frac{n\sqrt{x}}{1+n^2x^2}$$ I know I should use the Lebesgue dominated convergence theorem but what should be the dominating ...
1
vote
0answers
29 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
1
vote
3answers
54 views

property of Lebesgue integral

If $f$ and $g$ are nonnegative Lebesgue measurable functions, then we know that $\int (f+g) d\lambda = \int f d \lambda + \int g d \lambda $. Given the difinition of integral of an arbitrary Lebesgue ...
1
vote
2answers
77 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
1
vote
1answer
55 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
1
vote
0answers
35 views

lebesgue integral of $f(x^n)$

I know that $f:[0,1]\to \mathbb{R}$ is continuous at $0$, and $f\in L_1([0,1])$. How can one prove that $f(x^n)\in L_1([0,1])$, for any $n\in \mathbb{N}$?
1
vote
1answer
24 views

For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$?

The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values ...
0
votes
1answer
50 views

f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
0
votes
1answer
47 views

Every Cauchy sequence in $C([0,1])$ in the $L^2$ norm is also Cauchy in the $L^1$ norm.

I am asked to show the following: Proposition. There is a unique injection $j: L^2([0, 1]) \hookrightarrow L^1([0,1])$ which continuously extends $Id: C([0, 1]) \to C([0, 1])$. Here $L^1([0, ...
0
votes
1answer
81 views

Prove the Countable additivity of Lebesgue Integral.

Let $E\subset\mathbb{R}$ a measurable subset, $f\in L^1(E)$ and $\{E_n\}$ a disjoint countable union of measurables sets such that $\bigcup E_n=E$. Show that $$ \int_Ef=\sum_{n=1}^\infty\int_{E_n} ...
4
votes
2answers
86 views

Show that exists a function not increasing $f:(a,b)\rightarrow\mathbb{R}$ that is continuous only over $(a,b)\setminus D$

Let $D$ a infinity countable subset of $(a,b)$. Show that exists a function not increasing $f:(a,b)\rightarrow\mathbb{R}$ that is continuous only over $(a,b)\setminus D$ This is an exercise of my ...
2
votes
2answers
236 views

Showing that function is not Lebesgue Integrable in $[0,1]$

Is an exercise of my course of Measure and Integration. Let $f:[0,1]\rightarrow\mathbb{R}$ such that: $$ f(x)= \left\{ \begin{array}{ll} x^2\sin(\pi/x^2) & \textrm{ if } 0<x\leq 1\\ 0 ...
0
votes
1answer
64 views

Measure Theory - condition for Integrability

A question from my homework: Let $f:X\to [0,\infty)$ be a measurable function w.r.t to the Lebesgue measure and the Borel Sigma Algebra. Show that $\int_Xf(x)\,d\mu < \infty$ iff ...
2
votes
1answer
103 views

Showing that the integral of $x^nf(x)=0$ where $f$ is Lebesgue Integrable.

I'm trying solve this but I'm not sure that is correct. The exercise is: If $f:[0,1]\rightarrow\overline{\mathbb{R}}$ is lebesgue integrable, show that: $$ ...
3
votes
1answer
68 views

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

in preparation for an exam I wanted to show that for $\mu(\Omega)=1$ and $h:\Omega \rightarrow [0,\infty]$ measurable the following inequality holds: $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq ...
6
votes
2answers
204 views

Invariance of the Lebesgue integral.

Problem Let $f\in L^1(\mathbb{R})$. Show that $\int_{\mathbb{R}}f(x)dx=\int_{\mathbb{R}}f(x-\frac{1}{x})dx$. Discussion I know the Lebesgue integral is translation invariant (as the Lebesgue measure ...
3
votes
4answers
174 views

Lebesgue integral and Cantor set

I need to evaluate the integral $\int_{[0,1]} f \; d\mu $ using Lebesgue integral when $d\mu$ is Borel measurement and $f$ is given by: $$ f(x) = \begin{cases}x &x\in C, \\ ...
2
votes
1answer
66 views

Bounding a maximal function (Estimating an integral)

Let $\mathcal{R}$ denote the set of all open rectangles in $\mathbb{R}^2$ with sides parallel to the coordinate axis. Given a function on $\mathbb{R}^2$, consider the maximal function: ...
2
votes
1answer
107 views

Hardy-Littlewood maximal function weak type estimate

Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$, $$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$ where $C_q$ ...
1
vote
1answer
34 views

Which integral is greater?

I'd like to find out which of these two is greater: For a Lebesgue measurable function $f:[0,1] \rightarrow [1, \infty)$ (1): $\int_0^1 f(x)log(f(x))dx$ (2): $\int_0^1f(y)dy\int_0^1log(f(w))dw$ ...
0
votes
1answer
51 views

If $f_n \rightarrow f$ a.e. on $A, \lambda(A) < \infty$ then $\lim_{n\rightarrow \infty} \int_A f_n(x) dx = \int_A f(x) dx$

I want to show that if $f_n \rightarrow f$ a.e. on $A, \lambda(A) < \infty$ then $\lim_{n\rightarrow \infty} \int_A f_n(x) dx = \int_A f(x) dx$. I've tried to prove this but my proof is sitting ...
1
vote
2answers
44 views

Fix some $\delta\in \mathbb R$ and let $f:[0,\infty)\rightarrow \mathbb R$ be given by the equation

$$f(x)=\frac{\sin(x^2)}{x}+\frac{\delta x}{1+x}$$ Show that, $\lim_{n\rightarrow\infty}\int_{0}^{a}f(nx)\ dx=a\delta$ for each $\ a>0$. My attempt: $\lim_{n\rightarrow\infty}\ f(nx)=\delta$ ...
1
vote
1answer
58 views

Lebesgue integrability of $\ f$ and $\ f^{-1}$

Suppose $\ f:\ X\rightarrow(0,\infty)$ is a measurable function. If $$\int_{X} f\ d\mu<\infty\ $$ and $$\int_{X} \dfrac1f\ d\mu<\infty $$ Show that $\mu(X)<\infty$.
1
vote
1answer
55 views

How to apply Fubini here

Let $f$ and $g$ be integrable functions on the measure space $(X, \mathcal M, \mu)$ with the property that $$ \mu(\{f > t\} \triangle \{g > t\}) = 0 $$ for $\lambda$-a.e. $t \in \mathbb R$. ...
1
vote
0answers
111 views

If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise

I'd like to show that for an integrable sequence of functions $f_n:X \rightarrow [0, \infty)$ with $\sup_{n\geq 1} \int_{X} f_n d\mu < \infty, f_n \rightarrow f$ pointwise a.e. for some function ...
4
votes
2answers
282 views

Need help badly: n-dimensional Lebesgue measure of a hyperplane is zero.

Let $\alpha \in \mathbb{R}$, $a \neq 0$, and $\mu \in \mathbb{R}^{n}$. Let $H$ be the hyperplane in $R^{n}$ given by $h = \{ x \in \mathbb{R}^{n} : \langle x-\mu , a \rangle = 0 \}$. Show that ...
1
vote
1answer
66 views

Show $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$

I have to show that $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$, for $(A, \textit{S}, m)$ a measure space and $f: A \rightarrow \mathbb{R}$ an integrable function. Let $A_n = ...
1
vote
1answer
89 views

Integral of product of two measurable functions

I have to show that for $\psi$ and $\phi$ two positive and measurable functions: $$\int_X \phi\psi \, d\mu \le \sqrt{\int_X \phi^2 \, d\mu} \sqrt{\int_X \psi^2 \, d\mu}$$ I know that for $f,g$ two ...
3
votes
2answers
123 views

Is this function Lebesgue integrable? [duplicate]

I have to decice if the following function is Lebesgue-integrable on $[0,1]$: $$g(x)=\frac{1}x\cos\left(\frac{1}x\right) $$ where $x\in[0,1]$. $g(x)$ is Lebesgue integrable if and only if the ...
1
vote
1answer
128 views

Double expected value

Let $m$ be a probability measure on $\mathbb{R}^n$, so that $m(\mathbb{R}^n) = 1$. Consider two measurable functions $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and $g : ...
1
vote
1answer
42 views

$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
3
votes
1answer
102 views

Evaluating a Gaussian Integral

How to prove that $$\int_{\mathbb{R}^N}e^{-\langle Ax,x\rangle}\operatorname{dm}(x)=\left(\frac{\pi^N}{\det A}\right)^{\frac{1}{2}}$$ Where $A:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a symmetric ...
1
vote
0answers
74 views

Show derivative of integral equals integral of partial derivative if M[0,1]-measurable

I am trying to determine a method of approaching the following: Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
-1
votes
2answers
93 views

prove $ e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds $ [duplicate]

please How can I prove that $$ e^{-bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds $$ f non-negative measurable function I would appreciate it enormously if ...
0
votes
1answer
56 views

prove that integral

prove that $$-\int_0^t \ sgn(f({s})) \ d{s}=\int_0^t \ sgn(-f({s}))\ d{s}+2\int_0^t 1_{f({s}) =0}d{s}$$ with $$ sgn(x) := \begin{cases} -1 & \text{if } x =< 0, \\ 1 & \text{if } x > ...
1
vote
0answers
181 views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
2
votes
2answers
65 views

How to compute the limit of the following integral?

Given $b > 0$, let $g(x)$ be a continuous function defined on $[-b, b$]. What is the following limit? $\lim_{N \rightarrow \infty} \frac{1}{\sqrt{N}} \int_{-b}^b e^{-\frac{Nx^{2}}{2}}g(x)\,dx $ ...
-6
votes
1answer
134 views

Convergency of the functions $ϕ_nf_n$ converge to $ϕf$ in $L_1(μ)$

Suppose that a sequence of $μ$-integrable functions $f_n$ converges to $f$ in $L_1(μ)$ and a sequence of $μ$-measurable functions $ϕ_n$ converges to$ϕ$ $μ$-a.e. and is uniformly bounded. Show that the ...
1
vote
1answer
67 views

Converging almost surely and value of the integral

Given a sequence of nonnegative functions $(f_n)_n$ converging almost surely (for the Lebesgue measure $d\mu$) to $f$. Assume that $\int f_n d\mu \rightarrow c < \infty$ as $n$ goes to infinity. ...
0
votes
1answer
643 views

Lebesgue integrability of continuous function in closed interval

I'm trying to show that a continuous function $f$ in $[a,b]$ is Lebesgue integrable, using approximation through step functions. It is pretty trivial to show using the connection between Riemann ...
2
votes
1answer
30 views

Showing $\lim_{n \to\infty} \hat{g}(n) \to 0$

Let $g \in L^{1}(\mathbb{T})$. Show that $|\hat{g}(n)| \leqslant ||g||_{1}$ for all $n \in\mathbb{Z}$ and $\displaystyle \lim_{|n|\to\infty} \hat{g}(n)\to 0$.
1
vote
1answer
61 views

Find Values for which a Lebesgue Integral Exists

I am posting here a problem from my homework. I am having trouble with a number of problems, but I think guidance on this one should help me grasp the general concept and complete some of the others. ...
1
vote
2answers
394 views

Is $(\log x)e^{-x}$ Lebesgue integrable?

Could anyone give me any little hints on how to show the following please? Is $(\log x)e^{-x}$ Lebesgue integrable on $(0, \infty)$ ? I cannot see how to do this. I've tried Comparison Test. I ...
2
votes
0answers
319 views

solving this integral using Lebesgue dominated convergence theorem

I'm tried without sucess to solve this integral: $$\lim_{n\to\infty}\int_{0}^{1}(1+x^2)^{-n}dx$$ I know that the why to solve it ,is by using the Lebesgue dominated convergence theorem.
1
vote
0answers
95 views

Infinite function on measure zero set

I am trying to prove that given a set $E$ of measure 0, and a function $f \equiv \infty$ on $E$, then $\displaystyle \int_{E}f = 0$. This would be easy if one is allowed to assume that $\infty \times ...
2
votes
1answer
178 views

Bounded measurable function and integral with charcteristic function

I have been struggling with the following for quite some time now. If anyone can give me some help, it will be much appreciated: Let $f$ bounded, measurable and $E$ be a set of finite measure. Let $A ...
0
votes
2answers
47 views

calculate $\int_Df(x,y)dxdy$ in these cases

$f(x,y)={1\over{x^2y}}$ With $D=${$ x\ge1,{1\over{x}}\le y\le x $ } $f(x,y) = e^{2x+y}$ With $D=${ $x\le a ,x+y\le a $ }
0
votes
2answers
85 views

Calculating double integrals

We have the function $f:\mathbb R\to\mathbb R$: $$f(x) =\begin{cases} y^{-2} & \text{if } 0\lt x\lt y \le 1 \\ -x^{-2} & \text{if } 0\lt y\lt x \le 1 \\ 0 & \text{otherwise } ...